Orthorhombic charge density wave on the tetragonal lattice of EuAl4

The incommensurate charge density wave of EuAl4 below T CDW = 145 K is found to possess orthorhombic symmetry, despite an average crystal structure that remains tetragonal in very good approximation. This finding has ramifications for the interpretation of all physical properties of EuAl4, in particular, its multiple magnetic transitions.

S3. t-plots. S1. X-ray diffraction at beamline P24 of DESY Single-crystal X-ray diffraction (SXRD) was measured at station EH2 of beamline P24 of PETRA-III extension at DESY in Hamburg, Germany, employing radiation of a wave length of λ P 24 = 0.50000Å. Diffracted X rays were detected by a Pilatus 1M CdTe detector. The temperature of the sample was regulated with a CRYOCOOL open-flow cryostat, employing helium as cryo gas. Table S1 shows the thermal history of the compound along with the observed phases. The SXRD data at 250 K were measured with a crystal-to-detector distance of D = 110 mm. For 70 and 20 K we have collected SXRD data for two positions of the detector: 2θ offset = 0 and 25 deg, along with a long crystal-to-detector distance of 260 mm, in order to resolve the superlattice reflections. Consequently, the low-temperature data sets contain fewer (main) reflections than the data set at 250 K does.
Crystal A of dimensions 0.1×0.02×0.06 mm 3 was selected for the SXRD experiment at beamline P24. Diffracted intensity was collected on the detector during rotation of the crystal by 0.1 deg and 0.1 second exposure time. Each run of data collection comprises a 10 times repeated measurement of 3640 frames, corresponding to a total rotation of the crystal by 364 deg, repeated 10 times. These data were binned to 364 frames of 1 deg of rotation and 10 seconds exposure time, using the SNBL toolbox (Dyadkin et al., 2016).

S2. Symmetry of the modulated CDW state
Data processing of the binned data sets (Section S1) was performed with the software suite EVAL15 (Schreurs et al., 2010), resulting in the lattice parameters and a list of intensities of Bragg reflections. At this point, no deviations from tetragonal symmetry could be detected.
Structure refinements have been performed with the software Jana2006 against the experimental data set (Petricek et al., 2014). The modulation function for displacement modulation, is defined as where we have used n max = 1 (first-order harmonics).
Structure refinements were performed with a series of structure models differing in their symmetries according to different superspace groups (Table S2). Orthorhombic symmetries were used under the assumption of twinning, thus always leading to a diffraction symmetry close to 4/mmm. Table S2 allows for several observations: without satellite reflections and modulation, the data are excellently fitted by the tetragonal structure model. The same basic structure is obtained for refinements in the orthorhombic superspace groups (Table   S4). The only deviation from tetragonal symmetry is in the values of the ADPs.
This can be understood from the fact that for tetragonal superspace groups the same symmetry operator, (m, 0) or (m, s)-where "s" represents a phase shift of the modulation wave by half a wavelength (van Smaalen, 2012)-, exists perpendicular to a as well as perpendicular to b, or it exists perpendicular to both a ± b. The poor fit to Immm(00σ)000 and the good fit to Immm(00σ)s00 show that the different symmetry operators are required perpendicular to a than perpendicular to b. A similar observation can be made for F mmm(00σ)s00 and the symmetry operators (m, s) and (m, 0) perpendicular to a ± b. Obviously, tetragonal groups always include the wrong phase relation between modulation waves on different atoms. While the basic structure remains tetragonal in good approximation for all symmetries (Table S4), the modulation amplitudes for the two orthorhombic models in Table S5 illustrate the essentially different symmetry restrictions for these models.
All parameters (lower R values for a smaller number of parameters) then point to the orthorhombic superspace group F mmm(00σ)s00 as the correct symmetry of EuAl 4 in its CDW state (Tables S2 and S3).  (5) Table S5. Amplitudes of the modulation functions of crystal A at 70 K for Immm(00σ)s00 and F mmm(00σ)s00. Refined values have been multiplied by the corresponding lattice parameter, in order to obtain values inÅ. The temperature dependence of the magnetic susceptibility is shown in Fig. S1 [Compare to Nakamura et al. (2015)].
Fig. S1. Temperature dependent magnetic susceptibility of EuAl 4 from 2.4 to 300 K. Data measured in fields of 0.1 T and 0.5 T.